Wigner crystallization of electrons in deep traps in a two-dimensional dielectric
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ONIC PROPERTIES OF SOLID
Wigner Crystallization of Electrons in Deep Traps in a TwoDimensional Dielectric S. S. Shaimeev* and V. A. Gritsenko Institute of Semiconductor Physics, Novosibirsk, 630090 Russia *email: [email protected] Received July 8, 2010
Abstract—A twodimensional model is used to examine the spatial distribution of electrons in deep traps in a twodimensional dielectric. When the trap concentration is much higher than the trapped electron concen tration, Coulomb repulsion leads to the formation of a twodimensional quasiperiodic hexagonal lattice of localized electrons (Wigner glass). DOI: 10.1134/S106377611102021X
Coulomb repulsion between free electrons leads to Wigner crystallization, which has been observed on liquid helium surface [1]. In [2–4], it was hypothe sized that Wigner crystallization of trapped carriers (electrons and/or holes) could occur in a dielectric with high neutral trap concentration because of Cou lomb repulsion, as predicted by the original Wigner model. In [2], holes localized in silicon nitride were assumed to form a quasiperiodic square lattice. To examine Wigner crystallization of trapped elec trons and/or holes, we consider amorphous silicon nitride as a model dielectric. Silicon nitride has mem ory: it can hold electrons and holes in traps for as long as about 10 years at T = 450 K [5]. Practical interest in electron and hole localization in amorphous silicon nitride is motivated by the development of terabit flash memory based on silicon nitride [6]. There are deep electron and hole traps (about 1.5 eV deep) in this compound. According to studies of charge transfer, the neutral trap concentration in Si3N4 is Nt ~ 1019– 1020 cm–3 [7–11], whereas the concentration of occu pied traps is much lower, nt ~ (2–6) × 1018 cm–3 [12, 13]. In [14], electrons were observed to spread in sili con nitride, driven by their own repulsive Coulomb field. In this study, a twodimensional model is used to perform numerical simulations of Wigner crystalliza tion in a dielectric with deep traps. We consider a twodimensional dielectric with concentration Ns of randomly distributed neutral traps as a model of disordered amorphous structure of a real dielectric. A fraction of the traps are randomly occu pied by electrons, with concentration ns. A trapped electron can be released with probability P by thermal ionization. The ith free electron moves in the plane with the drift velocity Vi = μFi determined by the mag nitude and direction of the electric field Fi generated by other (both free and bound) electrons (μ is the elec
tron mobility). An electron that passes by at a distance from a neutral trap shorter than a certain ls is captured by the trap. Since only a finite number of traps can be used in numerical simulations, boundary conditions should be set with particular care. In our study, the system was a square with side L, and cyclic boundary conditions were imposed to avoid electric field distortion at its boundaries; i.e., simulations were performed for an infinite num
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